Separate the target population, frame, selection and response processes.
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Try
Suppose a council wants to know whether adult residents support a new bus timetable. An online link receives many responses from regular riders. Before calculating a percentage, draw three nested groups: all adult residents, people reachable through the chosen contact list, and people who actually respond. Which group does the observed percentage describe directly?
Predict
Predict whether doubling the number of volunteers removes an underrepresentation of people without internet access. More observations can reduce random fluctuation within the responding group while leaving the selection mechanism unchanged. Separate “many replies” from “a sample selected to represent the target.”
Experiment
Work with an original synthetic frame of 600 listed residents. A simple random sample selects 60 distinct names without replacement. Each listed resident has inclusion probability 60/600=0.1. Imagine that only 39 respond: the response rate is 39/60=0.65, not 39/600.
Draw two arrows, frame to selected sample and selected sample to respondents. Label random selection on the first and response behavior on the second. Now add 100 eligible adults absent from the list. They have zero inclusion probability through this frame. Recalculate neither their opinions nor a corrected support rate: those values are unknown. Identify what additional collection would be needed.
Understand
In a simple random sample of a fixed size, every subset of that size has equal selection probability. Giving individuals equal probability is useful but does not alone define an SRS: selecting an entire randomly chosen classroom gives students equal probabilities under some designs while permitting only particular groups.
Stratified sampling divides the population into nonoverlapping groups and samples within every group. It can ensure representation and improve precision when the strata relate to the response. A cluster sample randomly selects some groups, then observes all units in them in a one-stage design. Cluster sampling may reduce travel costs, but units within a cluster can be similar. Treating all their responses as independent may overstate precision.
Systematic sampling selects every kth unit after a random start. It can work well, but a periodic ordering aligned with k can distort the result. Convenience sampling chooses easy-to-reach units. Voluntary response lets people select themselves. Neither automatically supports the generalization justified by a probability design.
Undercoverage occurs when eligible units are absent from the frame. Nonresponse occurs when selected units do not provide data. Response bias concerns how answers or measurements differ systematically from the intended quantity, perhaps because of leading wording or interviewer effects. These are distinct mechanisms and need distinct repairs.
A larger random sample usually reduces sampling variability under its design. It does not automatically repair any of these systematic problems. Weighting can adjust documented representation differences under assumptions, but cannot magically recover unmeasured opinions or eliminate unknown selection effects.
Apply
Write a sampling plan for the council that names a frame, selection rule and follow-up process. Contrast a stratified sample from every neighborhood with a cluster sample of selected blocks. Explain both the practical cost and the statistical tradeoff. Do not describe “random” as simply choosing names without thinking: specify a reproducible chance mechanism.
Continue
Sampling governs population generalization. The next lesson introduces random assignment, a different use of chance that helps identify treatment effects within a study.
Apply · explain · revise
Practice and reasoning
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A simple random sample of 60 distinct people is drawn from 600 listed people. What is each listed person’s inclusion probability?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Of 160 selected people, 104 respond. What fraction responded?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A school randomly selects students within every grade. This is:
A biased web poll grows from 500 to 50,000 volunteers. Which statement is justified?
Describe a cluster sample of apartment residents and contrast it with sampling within every building.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A city surveys voters using landline numbers, then reports attitudes of all adult residents. Identify two distinct selection problems and a repair.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
